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   "source": [
    "# p19: Second order Wave Equation on Chebyshev Grid "
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We solve\n",
    "$$\n",
    "u_{tt} = u_{xx}+u_{yy}, \\qquad -1 < x,y < 1,   \\qquad t > 0\n",
    "$$\n",
    "with\n",
    "$$\n",
    "u = 0 \\qquad \\mbox{on the boundary}\n",
    "$$\n",
    "and initial condition\n",
    "$$\n",
    "u(x,y,0) = e^{-40((x-0.4)^2 + y^2)}, \\qquad u_{t}(x,y,0) = 0\n",
    "$$"
   ]
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  {
   "cell_type": "code",
   "execution_count": 3,
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   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "from chebfftPy import chebfft\n",
    "from numpy import arange,cos,zeros,round,exp,pi\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib.collections import LineCollection\n",
    "from matplotlib.pyplot import figure"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
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   ],
   "source": [
    "# Time-stepping by Leap Frog Formula:\n",
    "N = 80; t = 0.0 ;x = cos(pi*arange(0,N+1)/N);dt = 8.0/(N**2);\n",
    "v = exp(-200*x**2); vold = exp(-200*(x-dt)**2);\n",
    "tmax = 4 ; tplot = 0.075;\n",
    "plotgap = int(round(tplot/dt)); dt = tplot/plotgap;\n",
    "nplots = int(round(tmax/tplot));\n",
    "plotdata = [];plotdata.append(list(zip(x,v)));\n",
    "tdata = zeros(nplots+1);\n",
    "for i in range(1,nplots):\n",
    "    for n in range(plotgap):\n",
    "        t = t + dt\n",
    "        w = chebfft(chebfft(v)); w[0] = 0.0 ; w[N] = 0.0; \n",
    "        vnew = 2*v - vold + dt**2*w; vold = v ; v = vnew;\n",
    "    plotdata.append(list(zip(x,v)));\n",
    "    tdata[i] = t;\n",
    "\n",
    "fig = figure(figsize=(10,8))\n",
    "ax = fig.gca(projection='3d')\n",
    "poly = LineCollection(plotdata)\n",
    "poly.set_alpha(0.5)\n",
    "ax.add_collection3d(poly, zs=tdata, zdir='y')\n",
    "ax.set_xlabel('X')\n",
    "ax.set_xlim3d(-1, 1)\n",
    "ax.set_ylabel('Y')\n",
    "ax.set_ylim3d(0, tmax)\n",
    "ax.set_zlabel('Z')\n",
    "ax.set_zlim3d(-2, 2)\n",
    "ax.view_init(70,-70)"
   ]
  }
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